A Beta Item Response Model for Continuous Bounded Responses

Abstract

An item response model is proposed for the analysis of continuous response formats in an item response theory (IRT) framework. With such formats, respondents are asked to report their response as a mark on a fixed-length graphical segment whose ends are labeled with extreme responses. An interpolation process is proposed as the response mechanism in this situation, from which the authors show that a beta distribution on the response naturally follows. The two natural parameters of the beta are expressed as monotonic functions of person and item distance on some latent continuum. It is shown that a logistic expected response function results from a simple choice for these monotonic functions. A joint maximum likelihood estimation procedure is presented that performs well in a simulation study. An application on mood items is also presented, with very good fit properties.